where \(\Delta _0\)  is the restriction of \(\Delta \)  on the unit sphere. Another important property is that
they satisfy an addition formula [1]}: for \(\xi \in {\mathbb {S}^{d-1}}\)  and \(\eta \in {\mathbb {S}^{d-1}}\) ,
\( \sum _{\ell =1}^{\dim {\mathcal {H}}_n^d} Y_\ell ^n (\xi ) Y_\ell ^n(\eta ) = Z_n^{\frac{d-2}{2}} ({\langle }\xi ,\eta {\rangle }), \qquad Z_n^{\lambda }(t) = \frac{n+{\lambda }}{{\lambda }} C_n^{\lambda }(t),\) 
